I own an Arietta 570 and Parkers, both Repros and originals...I would choose the Parker Repro--although in 20-gauge it will a probably get a little heavy as a carry gun. For that reason, I'd recommend seeking out a 28-gauge Parker Repro instead--you'll pay more, but it will pay dividends down the road.
From what I've read and understand, Tom Skeuse, the originator of Parker Reproductions had intended that the Parker Reproductions be simply called Parkers, but Remington Arms owned the name and licensing and they (Remington) wouldn't allow Skeuse to call them Parkers--thus we have Parker Reproductions.
To my way of thinking, a Parker Reproduction is as much a Parker as a Galazan built Parker is a Parker--or a Remington Arms Ilion-built Remington Parker is a Parker. Along these same lines a Parker Repro is as much a Parker as a CSMC Fox is an A. H. Fox or a Ithaca Classic Doubles is an Ithaca.
